Fast iterative solvers for boundary value problems on a local spherical region

Abstract

Boundary value problems on local spherical regions arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Meshless methods using radial basis functions provide a simple way to construct numerical solutions with high accuracy. However, the linear systems arising from these methods are usually ill-conditioned, which poses a challenge for iterative solvers. We construct preconditioners based on an additive Schwarz method to accelerate the solution process for solving boundary value problems on local spherical regions. References

• D. Crowdy. Point vortex motion on the surface of a sphere with impenetrable boundaries. Physics of Fluids, 18:036602 (2006). doi:10.1063/1.2183627.
• A. E. Gill. Atmosphere-Ocean Dynamics, International Geophysics Series Volume 30. Academic, New York (1982).
• R. Kidambi and P. K. Newton. Point vortex motion on a sphere with solid boundaries. Physics of Fluids, 12:581 (2000). doi:10.1063/1.870263.
• Q. T. Le Gia, I. H. Sloan, and T. Tran. Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere. Math. Comp., 78:79--101 (2009). doi:10.1090/S0025-5718-08-02150-9.
• C. Muller. Spherical Harmonics, Vol. 17 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1966).
• M. V. Nezlin. Some remarks on coherent structures out of chaos in planetary atmospheres and oceans. Chaos, 4:109--111 (1994). doi:10.1063/1.165997.
• T. Tran, Q. T. Le Gia, I. H. Sloan, and E. P. Stephan. Preconditioners for pseudodifferential equations on the sphere with radial basis functions. Numer. Math., 115:141--163 (2009). doi:10.1007/s00211-009-0269-8.